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Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSE-Ecublens, 1015 Lausanne, Switzerland
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ABSTRACT |
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Abstract
Introduction
Methods
Results
Discussion
References
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Vasomotion has been studied on segments of rat mesenteric and femoral arteries perfused in vitro. We have investigated 1) the effect of perfusion flow on the characteristics of vasomotion and 2) the nature and patterns of vasomotion. We have found that perfusion flow is not a control parameter that contributes to the genesis of vasomotion but that it affects, in most cases only slightly, the frequency and amplitude of vasomotion. We have found evidence that vasomotion is low-dimensional chaotic. The correlation dimension ranged between 2 and 4, and the average Lyapunov's coefficient was ~0.1. A great variety of vasomotion patterns was observed with features that are typical of nonlinear deterministic systems: regular and irregular vasomotion, quasiperiodicity, period doubling and higher-order periods, intermittency, mixed modes, and bursting activity. Vasomotion patterns appeared occasionally to be highly sensitive to perturbations in perfusion flow, which also supported the existence of nonlinear dynamics. Finally, entrainment (phase locking) was observed when arteries were perfused with oscillatory flow with frequency in the neighborhood of the frequency of vasomotion.
chaos; vasomotion patterns; fractal dimension; rat arteries
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INTRODUCTION |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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ARTERIES OFTEN EXHIBIT spontaneous, rhythmic activity, which is manifested by low-frequency oscillations in vessel caliber, a phenomenon termed vasomotion. Vasomotion is predominant in microcirculation (6, 16, 20), but recently it has been shown to exist in large muscular arteries (14, 21, 23). In small arteries and arterioles, vasomotion influences perfusion (24), enhances filtration through the wall and lymphatic drainage (15), and may represent some form of homeostasis. In large arteries, the physiological significance of vasomotion remains obscure, although it has been demonstrated that vasomotion affects considerably the elastic properties of conduit vessels (26), and vasomotion in the coronary arteries has been suggested as an initiating factor of coronary spasm (23).
The mechanisms contributing to vasomotion have yet to be defined. Vasomotion has been closely linked to the myogenic mechanism (2, 5, 27). A number of studies (7, 9, 18) claim that vasomotion results from instabilities in the system that controls intracellular calcium. Mechanical factors such as wall stress and intimal shear appear to modulate vasomotion (3, 4, 11, 19, 20), but it has been suggested that these mechanical factors are not control variables of the system that generates vasomotion (13). The plurality of experimental patterns observed, sensibility to initial conditions, and nonlinear (fractal) analysis of vasomotion signals suggest that vasomotion is a chaotic process (9).
The objectives of the present study are 1) to examine the effect of flow on the characteristics of spontaneous vasomotion and determine whether flow is one of the control variables of the system that generates vasomotion, and 2) to assess the nature of vasomotion, examine its patterns, and determine whether low-dimensional deterministic chaos is involved.
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METHODS |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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Experimental Procedures
The effects of pressure and flow on the characteristics of arterial vasomotion were investigated in vitro on isolated rat mesenteric and femoral arteries. Mesenteric segments, typically 6 mm in length, were taken from the midportion of the fourth-order mesenteric branch (external diameter
0.4 mm and length 12 mm, in situ). Femoral
segments, typically 8 mm in length, were taken from the midportion of
the femoral artery (external diameter 0.7 mm and length 30 mm, in
situ). The excised arterial segments were mounted on microcannulas and
perfused with Tyrode solution (Living Systems Instrumentation,
Burlington, VT) using techniques similar to the work of Achakri et al.
(3). Vasomotion was induced and maintained by adding a constrictor
agonist in the superfusion solution (1 µM norepinephrine).
Endothelial function was checked in arteries by means of
acetylcholine-induced vasodilation. The experiments were terminated in
cases with no response to acetylcholine. The external diameter was
measured continuously using videomicroscopy, and pressure was measured
upstream and downstream of the arterial segment by means of a perfusion
pressure monitor. Flow was delivered by a syringe infusion pump
controlled by a personal computer. The temperature in the vessel
chamber (37°C) was controlled and monitored continuously. To study
the effect of flow, perfusion pressure was kept constant at 90 mmHg and
flow was varied between 100 and 800 µl/min in a stepwise manner. The
pressure drop across the test section at the highest flow (800 µl/min) was typically in the order of 10 mmHg. Changes in pressure of
10 mmHg or less do not induce significant myogenic responses, as we
have shown in earlier studies on the same arteries under similar
experimental conditions (3).
Theoretical Analysis
The correlation dimension of the vasomotion time series was estimated using the method of Grassberger and Procaccia (8). In brief, this method was implemented as follows. On the basis of the diameter time-series data, we constructed a set of m-dimensional vectors Xi = {di(t), di(t +
), ..., di[t + (m 
1)]}, where d is
diameter, t is time,
m is the embedding dimension, and is a time delay. We then calculated the correlation statistic
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(1) |
is the Heavyside function. The above function counts the number of
vector pairs
(Xi,
Xj) that lie
in the m-dimensional phase space and
at a distance shorter than r from each
other. Even the time series results from a deterministic system;
Grassberger and Procaccia (8) have shown that
C(r) is proportional to
r
, where the
exponent 
is in general nonintegral and gives an indication of the
fractal dimension of the time series. Therefore, the correlation
dimension can be obtained by plotting
C(r)
vs. r in logarithmic scale and taking
the slope of the graph as r tends to
zero and m tends to infinity, provided
that the embedding dimension is sufficiently large.
The choice of embedding dimension and time delay is critical for correct estimation of the correlation dimension (9, 10). In our analysis we used a time delay equal to 0.3 of the basic period of vasomotion and an embedding dimension of 8, which satisfies the Takens criterion (25). The Grassberger and Procaccia method (8) was tested first against time series from known chaotic systems (i.e., Lorenz and Henon attractors). The analysis was performed on 800-s segments of continuous recording sampled at 2.6 Hz. Similar analysis has been applied to vasomotion signals by Yamashiro et al. (28) and Griffith and Edwards (13).
The largest Lyapunov's exponent was estimated using the method of Rosenstein et al. (22).
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RESULTS |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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Effect of Flow
A typical result of an experiment on a femoral artery is presented in Fig. 1A, which shows the diameter at different perfusion flows and constant intraluminal pressure (90 mmHg). Flow ranged between 100 and 800 µl/min. At 800 µl/min the shear stress was ~35 and 15 dyn/cm2 for mesenteric and femoral arteries, respectively. Figure 1A shows that vasomotion amplitude and frequency depend on perfusion flow. Vasomotion frequency is plotted as a function of perfusion flow for mesenteric and femoral arteries in Fig. 1B. The frequency (f) decreased monotonically in both artery groups with an increase in perfusion flow (
),
although the absolute value of frequency differed between the two
groups. Linear regression yielded
f = 0.17 + 7.4 × 10
5
and f = 0.37 + 1.07 × 104
for femoral and mesenteric arteries, respectively.
The amplitude (A) of vasomotion decreased and was severely attenuated
at high flows in femoral arteries (Fig.
1C). Linear regression yielded A = 5.8 + 0.0054 and A = 8.1 + 0.0031 for femoral and mesenteric arteries,
respectively. The slopes of the linear regression curves were
significant.
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Vasomotion Patterns
Regular and irregular vasomotion. Vasomotion in the rat mesenteric artery exhibited a large variety of patterns. Two extreme cases are presented in Fig. 2. Figure 2A shows a typical case of regular vasomotion. The frequency spectrum exhibits a single peak at f = 0.32 Hz (Fig. 2C). Figure 2B shows a case of extremely irregular vasomotion, including a multitude of frequencies and large-scale variations in vessel diameter. The power spectrum is widely distributed over the range of 0-0.4 Hz (Fig. 2D). Such erratic vasomotion signals have been reported earlier in vivo and in vitro in both small and large vessels (9, 21).
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Specific patterns: routes to chaos. There is a certain parallelism between the vasomotion patterns observed in our vitro experiments and the behavior of nonlinear deterministic systems. Griffith (9) drew similar analogies in his review article. Nonlinear systems exhibit certain well-defined patterns when going from regular oscillatory behavior to chaos. These patterns signify transitional stages in the evolution toward a more complex nonlinear behavior and sometimes are called "routes to chaos." Typical cases of such specific patterns are shown in Fig. 3 and are classified as follows.
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Fractal Analysis and Chaos
We have calculated the correlation dimension () of the vasomotion
time series to determine 1) the
minimum number of independent control variables involved and
2) whether mechanical factors such as the intimal shear stress may influence the apparent complexity of
the system. Figure
4A shows
the estimated correlation dimension as a function of perfusion flow.
The values lie in the range of 2 < < 4, and there was no
statistically significant dependence on perfusion flow. We therefore
conclude that there is a minimum number of four control variables in
the vasomotion-generating system and that the level of flow does not
alter the fractal dimension of the system. Lyapunov's exponent was
estimated to be positive and typically around 0.1 (Fig.
4B).
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Sensitivity to External Perturbations
Although flow seems to be an external factor influencing the amplitude and frequency of vasomotion (Fig. 1), in certain cases changes in flow resulted in large-scale changes in vasomotion characteristics. Figure 5 shows an example in which step changes in flow induced spectacular and reversible changes in oscillatory modes. At a baseline value of 100 µl/min, high-frequency (f = 0.36 Hz), low-amplitude (
d = 10 µm) vasomotion prevails,
with characteristics similar to the high-frequency component of
vasomotion in Fig. 3A. When flow
increased to higher levels, vasomotion changed to a low-frequency mode
(f = 0.025 Hz) with a 10-fold increase in amplitude (d = 120 µm) and
characteristics comparable to the low-frequency component of vasomotion
shown in Fig. 3A.
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Entrainment
A periodic perturbation, such as oscillatory flow, was able to lock or entrain vasomotion. An example of entrainment is shown in Fig. 6, in which perfusion flow was changed from constant flow (400 µl/min) to oscillatory flow for a certain period of time (~100 s). The frequency of oscillatory flow was set at different levels ranging from 0.07 to 0.15 Hz. At constant flow, the vasomotion frequency was 0.16 Hz. We observe that in the presence of oscillatory flow, vasomotion phase locks onto the frequency of the oscillatory flow. The change in frequency is accompanied in general by an augmentation in vasomotion amplitude (see Fig. 6, top) that, however, becomes insignificant when the frequency of oscillatory flow approaches the frequency of spontaneous vasomotion (Fig. 6, bottom). No phase locking was observed for oscillatory flow with frequencies <0.07 Hz, which is approximately one-half the frequency of spontaneous vasomotion.
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DISCUSSION |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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We have studied vasomotion in rat mesenteric and femoral arteries in vitro, and we have found good evidence that vasomotion is low-dimensional chaotic. Mechanical forces, such as intimal shear, are not control variables of the system that generates vasomotion, and they influence slightly the characteristics of regular high-frequency vasomotion. Vasomotion patterns exhibit characteristics of nonlinear dynamic systems and show large-scale sensitivity to external perturbations.
Regular vasomotion patterns are characterized by a high-frequency (a period of 3-10 s) and a low-frequency component (a period of 40-60 s). These two regular oscillatory modes can in certain cases coexist. Griffith and Edwards (12) provided evidence that the slow oscillatory component is due to calcium release and uptake from the sarcoplasmic reticulum in smooth muscle cells, whereas the fast component is due to membrane oscillators. The interaction of the two subsystems can turn unstable, leading to chaos (9).
In all cases studied, the correlation dimension varied between 2 and 4. These results are consistent with the findings of Griffith and Edwards (13), who have showed that the fractal dimension of histamine-induced vasomotion in the rabbit ear artery was typically in the range of 2-3 and occasionally between 3 and 4. The fractal dimension was found not to depend on the level of flow and was not altered when the endothelium was functionally blocked by administration of NG-nitro-L-arginine methyl ester. In earlier studies we calculated the correlation dimension of vasomotion time series measured in human radial artery in vivo (1). The correlation dimension ranged between 2 and 3, which suggests that the complexity, and probably nature, of the vasomotion observed in vitro is similar to that in vivo.
The largest Lyapunov's exponent was positive, supporting the hypothesis that vasomotion results from a chaotic system. Typical features of a chaotic system are high sensibility to initial conditions and the possibility of large-scale changes in the trajectory (i.e., oscillatory patterns) following rather small perturbations by external factors. Therefore, the large spectrum of oscillatory modes and patterns as well as the extreme differences in vasomotion patterns observed for different levels of flow may be attributed to the chaotic nature of vasomotion (Fig. 5).
The results shown in Fig. 5 are not meant to contradict the dependence of vasomotion frequency and amplitude on flow shown in Fig. 1, because the results of Fig. 1 were primarily drawn from the analysis of regular, high-frequency vasomotion signals. Figure 5 shows nicely the rich and, to a large extent, unpredictable response of a nonlinear system to small external perturbations. When the weak dependence of mean diameter on mean flow is taken into account, a linear system is not capable of producing a response to changes in flow such as those shown in Fig. 5. The unpredictability of responses to external mechanical and pharmacological stimuli has been reported earlier for in vivo and in vitro preparations. For example, Griffith and Edwards (13) studied vasomotion induced by histamine in isolated rabbit ear resistance arteries and have shown that increases in flow provoked oscillations in some preparations, whereas in others they attenuated and ultimately abolished periodic oscillations. The possibility of "butterfly effects" or the application of nonlinear techniques for controlling chaos in vasomotion has been discussed extensively by Griffith (9).
We have computed the Lyapunov's exponent and found consistently positive values in the neighborhood of 0.1. Lyapunov's exponent characterizes the divergence over time of the phase-space trajectories. A positive Lyapunov's exponent means that divergence will grow in time, which points toward high sensitivity in initial conditions and chaos. Similar values for Lyapunov's exponents were estimated for vasomotion in rabbit ear arteries (10). Therefore, one of the contributions of the present study is the reconfirmation and generalization of earlier estimates of the correlation dimension and Lyapunov's exponent (9-13, 28) in different arteries and species.
Phase locking or entrainment may be another indication for the existence of nonlinear dynamic mechanisms controlling vasomotion. Phase locking between weakly coupled nonlinear oscillators is a well-known phenomenon occurring in biology (e.g., synchronized flashing of large populations of fireflies and chirping of crickets). In vivo studies have also revealed a high degree of synchronization in vasomotion between adjacent vessels in microcirculatory networks (17) as well as in large muscular arteries of the human arm (21).
The nonlinear dynamics involved in the development of vascular tone may have considerable implications in the physiology and pathology of the arterial system. Vasomotion is the manifestation of a series of complex events at the cellular (microscopic) and tissue (macroscopic) levels along which important control mechanisms (myogenic, nitric oxide synthase) are implicated. An understanding of vasomotion will enhance our basic knowledge of the control mechanisms and functional properties of arteries, and therefore we believe that vasomotion merits further investigation.
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FOOTNOTES |
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Address for reprint requests: N. Stergiopulos, Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSE-Ecublens, 1015 Lausanne, Switzerland.
Received 5 May 1997; accepted in final form 9 February 1998.
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REFERENCES |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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indicate successive periods. E:
intermittency. F: mixed modes.
G: bursting activity.
